| L(s) = 1 | + 3-s − 5·5-s − 2·7-s − 4·9-s + 8·11-s − 4·13-s − 5·15-s − 17-s + 10·19-s − 2·21-s + 4·23-s + 15·25-s − 5·27-s + 11·29-s − 3·31-s + 8·33-s + 10·35-s − 5·37-s − 4·39-s + 16·41-s + 31·43-s + 20·45-s + 5·47-s − 12·49-s − 51-s + 8·53-s − 40·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.23·5-s − 0.755·7-s − 4/3·9-s + 2.41·11-s − 1.10·13-s − 1.29·15-s − 0.242·17-s + 2.29·19-s − 0.436·21-s + 0.834·23-s + 3·25-s − 0.962·27-s + 2.04·29-s − 0.538·31-s + 1.39·33-s + 1.69·35-s − 0.821·37-s − 0.640·39-s + 2.49·41-s + 4.72·43-s + 2.98·45-s + 0.729·47-s − 1.71·49-s − 0.140·51-s + 1.09·53-s − 5.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.684686267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.684686267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{5} \) | |
| 37 | $C_1$ | \( ( 1 + T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 - T + 5 T^{2} - 4 T^{3} + 4 p T^{4} - 14 T^{5} + 4 p^{2} T^{6} - 4 p^{2} T^{7} + 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | 5.3.ab_f_ae_m_ao |
| 7 | $C_2 \wr S_5$ | \( 1 + 2 T + 16 T^{2} + 40 T^{3} + 191 T^{4} + 312 T^{5} + 191 p T^{6} + 40 p^{2} T^{7} + 16 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | 5.7.c_q_bo_hj_ma |
| 11 | $C_2 \wr S_5$ | \( 1 - 8 T + 6 p T^{2} - 324 T^{3} + 1541 T^{4} - 5224 T^{5} + 1541 p T^{6} - 324 p^{2} T^{7} + 6 p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.11.ai_co_amm_chh_ahsy |
| 13 | $C_2 \wr S_5$ | \( 1 + 4 T + 41 T^{2} + 128 T^{3} + 802 T^{4} + 2040 T^{5} + 802 p T^{6} + 128 p^{2} T^{7} + 41 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.13.e_bp_ey_bew_dam |
| 17 | $C_2 \wr S_5$ | \( 1 + T + 35 T^{2} + 48 T^{3} + 636 T^{4} + 926 T^{5} + 636 p T^{6} + 48 p^{2} T^{7} + 35 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | 5.17.b_bj_bw_ym_bjq |
| 19 | $C_2 \wr S_5$ | \( 1 - 10 T + 99 T^{2} - 612 T^{3} + 3590 T^{4} - 16068 T^{5} + 3590 p T^{6} - 612 p^{2} T^{7} + 99 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | 5.19.ak_dv_axo_fic_axua |
| 23 | $C_2 \wr S_5$ | \( 1 - 4 T + 63 T^{2} - 224 T^{3} + 2278 T^{4} - 6328 T^{5} + 2278 p T^{6} - 224 p^{2} T^{7} + 63 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.23.ae_cl_aiq_djq_ajjk |
| 29 | $C_2 \wr S_5$ | \( 1 - 11 T + 117 T^{2} - 860 T^{3} + 5686 T^{4} - 31586 T^{5} + 5686 p T^{6} - 860 p^{2} T^{7} + 117 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \) | 5.29.al_en_abhc_iks_abusw |
| 31 | $C_2 \wr S_5$ | \( 1 + 3 T + 59 T^{2} + 156 T^{3} + 2622 T^{4} + 8642 T^{5} + 2622 p T^{6} + 156 p^{2} T^{7} + 59 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | 5.31.d_ch_ga_dww_muk |
| 41 | $C_2 \wr S_5$ | \( 1 - 16 T + 162 T^{2} - 1190 T^{3} + 9357 T^{4} - 62804 T^{5} + 9357 p T^{6} - 1190 p^{2} T^{7} + 162 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.41.aq_gg_abtu_nvx_adoxo |
| 43 | $C_2 \wr S_5$ | \( 1 - 31 T + 509 T^{2} - 5716 T^{3} + 49760 T^{4} - 355946 T^{5} + 49760 p T^{6} - 5716 p^{2} T^{7} + 509 p^{3} T^{8} - 31 p^{4} T^{9} + p^{5} T^{10} \) | 5.43.abf_tp_ailw_cvpw_augog |
| 47 | $C_2 \wr S_5$ | \( 1 - 5 T + 71 T^{2} - 220 T^{3} + 4442 T^{4} - 23534 T^{5} + 4442 p T^{6} - 220 p^{2} T^{7} + 71 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.47.af_ct_aim_gow_abive |
| 53 | $C_2 \wr S_5$ | \( 1 - 8 T + 186 T^{2} - 1054 T^{3} + 16293 T^{4} - 74884 T^{5} + 16293 p T^{6} - 1054 p^{2} T^{7} + 186 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) | 5.53.ai_he_aboo_ycr_aegue |
| 59 | $C_2 \wr S_5$ | \( 1 - 24 T + 451 T^{2} - 6036 T^{3} + 62790 T^{4} - 545288 T^{5} + 62790 p T^{6} - 6036 p^{2} T^{7} + 451 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \) | 5.59.ay_rj_aiye_doxa_abfaqq |
| 61 | $C_2 \wr S_5$ | \( 1 + 15 T + 159 T^{2} + 2160 T^{3} + 19988 T^{4} + 139282 T^{5} + 19988 p T^{6} + 2160 p^{2} T^{7} + 159 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \) | 5.61.p_gd_dfc_bdou_hyba |
| 67 | $C_2 \wr S_5$ | \( 1 - 12 T + 147 T^{2} - 12 T^{3} - 6898 T^{4} + 125200 T^{5} - 6898 p T^{6} - 12 p^{2} T^{7} + 147 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) | 5.67.am_fr_am_akfi_hdfk |
| 71 | $C_2 \wr S_5$ | \( 1 - 5 T + 277 T^{2} - 1308 T^{3} + 35204 T^{4} - 134302 T^{5} + 35204 p T^{6} - 1308 p^{2} T^{7} + 277 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.71.af_kr_abyi_caca_ahqrm |
| 73 | $C_2 \wr S_5$ | \( 1 - 5 T + 287 T^{2} - 1064 T^{3} + 37048 T^{4} - 105670 T^{5} + 37048 p T^{6} - 1064 p^{2} T^{7} + 287 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | 5.73.af_lb_aboy_ccuy_agaig |
| 79 | $C_2 \wr S_5$ | \( 1 - 4 T + 223 T^{2} - 1124 T^{3} + 22318 T^{4} - 128176 T^{5} + 22318 p T^{6} - 1124 p^{2} T^{7} + 223 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) | 5.79.ae_ip_abrg_bhak_ahhpw |
| 83 | $C_2 \wr S_5$ | \( 1 - 27 T + 527 T^{2} - 6556 T^{3} + 72526 T^{4} - 655314 T^{5} + 72526 p T^{6} - 6556 p^{2} T^{7} + 527 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} \) | 5.83.abb_uh_ajse_edhm_ablhkk |
| 89 | $C_2 \wr S_5$ | \( 1 - 16 T + 317 T^{2} - 2896 T^{3} + 34778 T^{4} - 252480 T^{5} + 34778 p T^{6} - 2896 p^{2} T^{7} + 317 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \) | 5.89.aq_mf_aehk_bzlq_aojmu |
| 97 | $C_2 \wr S_5$ | \( 1 + 19 T + 539 T^{2} + 7208 T^{3} + 109668 T^{4} + 1040842 T^{5} + 109668 p T^{6} + 7208 p^{2} T^{7} + 539 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \) | 5.97.t_ut_krg_ggga_chfsk |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.61520188138704165059070455425, −5.43846033899373490044443269642, −5.32110842242707418188179985644, −5.20812599501647194822624157840, −5.05224348952274708317092173593, −4.72115002948596467099441668750, −4.46626610180775852602762385405, −4.40346158587637574309514131906, −4.16160306003463538253052518056, −3.97362997310010208999229781106, −3.91331946867475961827841439269, −3.44510229527239310481063496448, −3.43915397723303425855806444210, −3.43187712721616190907506368693, −3.23135652420503878762427817298, −2.69811154423413675932036377434, −2.65851498759014490575243649563, −2.43637978114955636769808807903, −2.41340336392409779298030448841, −2.05995521917391481419054946279, −1.41554838357700932132822508808, −1.04745989450988111344053866161, −0.942185227561967910124458952638, −0.66223202479245627673747187551, −0.50520690644701106109755732442,
0.50520690644701106109755732442, 0.66223202479245627673747187551, 0.942185227561967910124458952638, 1.04745989450988111344053866161, 1.41554838357700932132822508808, 2.05995521917391481419054946279, 2.41340336392409779298030448841, 2.43637978114955636769808807903, 2.65851498759014490575243649563, 2.69811154423413675932036377434, 3.23135652420503878762427817298, 3.43187712721616190907506368693, 3.43915397723303425855806444210, 3.44510229527239310481063496448, 3.91331946867475961827841439269, 3.97362997310010208999229781106, 4.16160306003463538253052518056, 4.40346158587637574309514131906, 4.46626610180775852602762385405, 4.72115002948596467099441668750, 5.05224348952274708317092173593, 5.20812599501647194822624157840, 5.32110842242707418188179985644, 5.43846033899373490044443269642, 5.61520188138704165059070455425
